On proving $a^3+b^3+c^3-3abc \geq 2\left({b+c\over 2}-a\right)^3$.

Question

This problem was a "warm-up" problem by the author. Note: $a, b, c$ are non-negative numbers. $$a^3+b^3+c^3-3abc \geq 2\left({b+c\over 2}-a\right)^3$$ I tried to remove the $2$ from ${b+c\over 2}$ and got this- $$ 4(a^3+b^3+c^3-3abc) \geq (b+c-2a)^3 $$ $$ \Rightarrow 2(a+b+c)((a-b)^2+(b-c)^2+(c-a)^2) \geq (b+c-2a)^3 $$ But couldn't take it any further, but it looks as if Hölder's Inequality may help. And also I don't think that it is a "warm-up" problem.
Any help will be appreciated.

Answer 1

Because $$a^3+b^3+c^3-3abc-2\left(\frac{b+c}{2}-a\right)^3$$ $$=\frac{3}{4}\Big[4a^3-4(b+c)a^2+2(b^2+c^2)a+b^3-b^2c-bc^2+c^3\Big]$$ $$=\frac{3}{4}\Big[a(2a-b-c)^2+a(b-c)^2+(b+c)(b-c)^2\Big]\geqslant 0.$$ The equality occurs for $$a(2a-b-c)^2=a(b-c)^2=(b+c)(b-c)^2=0.$$

1. $a=0$.

Thus, $(b+c)(b-c)^2=0,$ which gives $b=c$.

2. $a\neq0$.

Thus, since $a(b-c)^2=0,$ we obtain $b=c$ and since $a(2a-b-c)^2=0,$

we obtain $a=b=c$.

Answer 2

If $b+c<2a$ then $$LHS \geq 0 \ge RHS.$$ If $b+c \geq 2a.$ We write inequality as $$4(a^3+b^3+c^3-3abc) \geq (b+c-2a)^3,$$ or $$4(a+b+c)(a^2+b^2+c^2-ab-bc-ca) \geq (b+c-2a)^3.$$ Because $a+b+c = 3a+(b+c-2a) \geq b+c-2a,$ so we will show that $$4(a^2+b^2+c^2-ab-bc-ca)\geq (b+c-2a)^2,$$ equivalent to $$3(b-c)^2 \geq 0.$$ Which is true. Equality holds when $ a=b=c$ or $ a=0,\,b=c.$

Note. We have $$a^3+b^3+c^3-3abc-{2\left(\frac{b+c}{2}-a\right)^3}$$ $$=(b-c)^2+\frac{3\,a}{2} \Big[(a-b)^2+(b-c)^2+(c-a)^2\Big] \geq 0.$$

Answer 3

My first SOS is same as Nguyen Huyen. Here is my second and third.

$$\displaystyle a^3+b^3+c^3-3abc-2\left(\frac{b+c}{2}-a\right)^3$$

$$\displaystyle=\frac34 \left( b-c \right) ^{2} \left( b+c \right) +\frac34 \left( a-b \right) ^{2}a+\frac14\, \left( a+b-2c \right) ^{2}a+\frac12\, \left(b+c-2a \right) ^{2}a$$ $$\displaystyle =\frac34\cdot {\frac { \left( 2\,{a}^{2}-2\,ac+{b}^{2}-{c}^{2} \right) ^{2}}{2 \,a+b+c}}+{\frac {3 a\left( a+b+c \right)\left( a-b \right) ^{2} }{ 2\,a+b+c}}$$

Answer 4

Using algebra.

Consider that we look for the minimum value of $$F=a^3+b^3+c^3-3abc -2\left({b+c\over 2}-a\right)^3$$ $$\frac{\partial F}{\partial a}=3 a^2+6 \left(\frac{b+c}{2}-a\right)^2-3 b c \tag 1$$ $$\frac{\partial F}{\partial b}=3b^2-3 \left(\frac{b+c}{2}-a\right)^2-3 a c\tag 2$$ $$\frac{\partial F}{\partial c}=3c^2-3 \left(\frac{b+c}{2}-a\right)^2-3 a b\tag 3$$

Using $(3)$ the only possible value of $c$ is $$c=\frac{1}{3} \left(2 \sqrt{4 a^2-a b+b^2}-2 a+b\right)\tag 4$$ Plug in $(2)$ and solve for $b$; the only possible solutions are $$b_1=a \qquad \text{and} \qquad b_2=\frac{\sqrt{6}-1}{2} a\tag 5$$ $$b=b_1\implies\frac{\partial F}{\partial a}=0$$

$$b=b_2\implies\frac{\partial F}{\partial a}=\frac{1}{12} \left(359-140 \sqrt{6}\right) a^2$$ which must be rejected.

So $b=a$ and $c=a$ then $a=b=c$ and $F=0\,\, \forall a >0$